Axioms for the category of Hilbert spaces

Hilbert spaces and their operators are the mathematical foundation of quantum mechanics. The problem of reconstructing this foundation from first principles has been open for nearly a century. What is mathematically special about Hilbert spaces and their operators? This paper gives a categorical axiomatization. Unlike previous partial results of this type, the axioms do not presuppose probabilities, complex amplitudes, or continuity and are not limited to finite dimension.